It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic closed-form formulas. And I remember reading somewhere that the roots of quintic equations can be expressed in terms of the hypergeometric function.

What is known, beyond Abel-Ruffini, about closed-form formulas for roots of polynomials? Does there exist a formula if we allow the use of additional special functions?

$\begingroup$Indeed the Galois group of $X^n-1$ is not only solvable, but also abelian.$\endgroup$
– JSchlatherFeb 1 '13 at 7:26

$\begingroup$I'm sorry. You are right. I was aiming at CM-fields and extensions thereof generated by values of $j$. I think it should be possible to construct a non-solvable extension generated by the value of $j$.$\endgroup$
– Hans GiebenrathFeb 1 '13 at 9:32

2 Answers
2

As you might already know, solutions to the quintic can be expressed in terms of either ${}_4 F_3$ hypergeometric functions or Jacobi theta functions. See King or Prasolov/Solovyev for details.

For polynomials of higher degree, there is also a general formula for the roots, due to Umemura. The formulae involve the multidimensional generalization of the Jacobi theta functions (the Riemann theta function), and are a bit unwieldy; see Umemura's paper if you want more details. See also this preprint for a solution of the reduced polynomial equation $x^n-x-\alpha=0$ in terms of hypergeometric functions.

general polynomial of degree 5 or higher have "no closed-form formula"

This is not exactly true, what it should be said is that "general algebraic equations of degree higher than 4 do not admit solutions by radicals" which means that they cannot be solved by operations implying combinations of ordinary additions, multiplications, divisions, raising to powers, root extractions...
On the other side it was shown by Hermite that fifth degree equations can be solved using the modular elliptic functions, which provide a generalization of the so called trigonometric solution of eqs. with degree lower than 5.
Higher order (than 5) algebraic equations can be solved by employing other forms of elliptic functions.